Silver and gold badges are always considered veteran-worthy, so you'll always be notified of those regardless of reputation or site.
On per-site metas, you're always considered a veteran; on all other sites (including Meta Stack Exchange), you're considered a veteran if you have the network association bonus rights, i.e. you have earned 200 rep somewhere in the past.
The following bronze badges are considered veteran-worthy:
- Nice Answer
- Nice Question
- Caucus
- Excavator
- Mortarboard
- Popular Question
- Proofreader
- Revival
- Synonymizer
- Vox Populi
- Announcer
- Precognitive
- Curious
- Tag badges
Any bronze badge not in this list is not veteran-worthy, i.e. won't cause a notification for veteran users.
Back before its retirement, the Tumbleweed badge was also included on the above list.